Question

If vectors s and b are orthogonal then,
||2s − 3b||^2 = 4||s||^2 + 9||b||^2.
True or False and explain.

I have no idea how to start this problem,

I know,

b dot s = 0
therefore,
sin(theta)||b||s|| = 0

Not sure how i can apply this to the problem though.

Answer 1

I would try use the fact that ||v||^2 = v dot v to expand both sides. See what happens

Answer 2

looks true to me, just use the fact that \(\large a^2 = a \bullet a\)

Answer 3

oh damn, I cant believe I missed that property ugh

Answer 4

apply the definition of distance and inner product
||a||^2 = (a,a)

Answer 5

so that ||2s-3b||^2 =(2s-3b, 2s-3b)

Answer 6

it's easy now, right? can you continue?

Answer 7

I dont think I follow actually ugh

Answer 8

ok,
((2s-3b),(2s-3b))
left distribute
=((2s-3b),2s)-((2s-3b),3b)

Answer 9

left distribute again
((2s,2s)-(3b,2s)-(2s,3b)+(3b,3b)

Answer 10

the 2middle term =0 because s \(\perp\)b
so that you just have (2s,2s)+(3b,3b)

Answer 11

by definition (2s,2s) =||2s||^2
and (3b,3b)=||3b||^2
done

Answer 12

one more thing, take constants out to get your form 4||s||^2+9||b||^2

Answer 13

wait I think I see,

||2s - 3b||^2 = (2s - 3b) ∙ (2s - 3b)
=
(2s - 3b) ∙ (2s - 3b) = 2s∙2s - 2s∙3b - 3b∙2s + 3b∙3b
=
2s∙2s - 2(2s∙3b) + 3b ∙ 3b

oh ok I get it thanks

Answer 14

if they weren't orthogonal -2(2s∙3b) =/= 0 therefore s and b must be orthogonal to be equal

Answer 15

Ugh I need to study the rules harder thanks loser for the assistance

Answer 16

I don't know why you use \(\bullet\) , it is not the correct notation. However, If you are familiar with this, it 's ok

Answer 17

I dont see how it is incorrect it is the dot product and that is a dot?

Answer 18

we use comma.

Answer 19

throughout my textbook it is a dot and in my notes so I dont know maybe your way is correct, does it really matter that much?

Answer 20

I don't know, when I took elementary linear algebra, my prof used dot, too. But when I took advance linear, the prof use comma and all reference books use comma. So that I use it only.

Answer 21

gtg, see you later.

Answer 22

That is pretty funny, I dont know what to tell you. Maybe it is just to separate the men from boys.

Answer 23

I am sorry for leaving before you finished. My food burned...... hehehe.. that's the reason.